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unit 7 Surface area and Volume 197

managers of companies that produce food products must decide how to package their goods, which is not as simple as you might think. many factors play into the decision of the size and shape of a food container, such as volume—or “How much will it hold?”—and surface area—or “How much will it cost to wrap the item this way?” the managers also consider how the individual containers will fit into larger boxes for shipping to stores. other considerations are the dimensions of grocery store shelves and whether or not the prod-uct is flat on at least one side and can be stacked. and managers never forget about their customers. they think about the appear-ance of the package and its practicality. they ask, “Will a customer be attracted to this?” and “is this container easy to carry?”

this unit is about the surface area and volume of solids. as you study each solid, you will recognize some of them as the shapes of certain food containers.

UNIT OBJECTIVES

► identify these figures and their parts: prisms, pyramids, cylinders, cones, and spheres.

► find the surface area and volume of a prism.

► find the surface area and volume of a pyramid.

► find the surface area and volume of a cylinder.

► find the surface area and volume of a cone.

► find the surface area and volume of a sphere.

► transform figures in three-dimensional space.

► describe how a solid of revolution is generated.

► identify planes of symmetry for solid figures.

UNIT 7 Surface Area and Volume

Copyright © 2006, K12 Inc. All rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the express prior written consent of K12 Inc.

Surface area and Volume 199

Surface Area and Volume

OBJECTIVES A company that manufactures baseballs wants to find the least expensive way to ship 1000 boxes of its product. There are many ways to decrease the cost of shipping. Two of them involve calculating the surface area and volume of the shipping boxes. If the company can increase the number of baseballs that fit into a shipping box, without increasing the cost of shipping the box, then the company will save money. Similarly, if the same number of baseballs can be shipped in a box with less surface area, the company will save money on the cost of the boxes. The company needs to maximize the volume of the shipping boxes while minimizing the surface area.

KEYWORDS cubesurface areasurface area to volume ratiovolume

Right Rectangular Prisms

Recall that the surface area of a solid is the sum of the areas of the outer surfaces of the figure. You can use the net of a right rectangular prism to develop a formula for finding its surface area. In the sketch below, the area of each face is written as a product of its dimensions (length, width, and height), with the sum written to the right. Notice that there are three pairs of congruent faces.

► Use formulas to calculate the surface areas of right rectangular prisms and cubes.

► Use formulas to calculate the volumes of right rectangular prisms and cubes.

► Find the surface area to volume ratios of figures.

► Solve problems using the ratio of surface area to volume.

► Determine the changes in perimeter, area, surface area, and volume in common geometric figures or solids when the value of one or more lengths has changed.


200 unit 7 Surface area and Volume

s

Formula for the Surface Area of a Right Rectangular Prism

The formula for the surface area S of a right rectangular prism of length l, width w, and height h is

S = 2lw + 2lh + 2wh

The calculator screen below shows how to find the surface area of the prism displayed in the previous sketch.

Recall that the volume of a solid is the number of unit cubes inside the figure. You can use transformations or isometric dots to draw the prism of the net in the previous sketch. The number of unit cubes that fills the figure is equal to the product of the length, width, and height of the prism.

Formula for the Volume of a Right Rectangular Prism

The formula for the volume V of a right rectangular prism of length l, width w, and height h is

V = lwh

The volume of the prism in the diagram above is 4·2·3, or 24 cm3.

Surface Area and Volume of CubesWhen every face of a right rectangular prism is a square, the prism is called a cube. Because a square has sides of equal length, we call each side s.

The area of every face of a cube is the same. That means it is possible to find a cube’s surface area by multiplying the area of just one face by 6.



6 cm 6 cm

4 cm

A = 24 cm2

P = 20 cmA = 48 cm2

P = 28 cm

8 cm

Formula for the Surface Area of a Cube

The formula for the surface area S of a cube of side s is S = 6s2

The calculator screen shows how to use this formula to find the surface area of a cube with a side length of 2 units.

By substituting s for l, w, and h, in the formula for the volume of a right rectangular prism, you can derive the formula for the volume a cube.

Formula for the Volume of a Cube

The formula for the volume V of a cube of side s is V = s3

To cube a number on a graphing calculator, use the caret key (^). The display shows that a cube with a side length of 2 units has a volume of 8 cubic units.

Changing Attributes of Figures Below, the length of a rectangle is doubled. Notice that when one dimension is doubled, the area doubles.



6 cm 12 cm

4 cm

A = 24 cm2

P = 20 cm8 cm

A = 96 cm2

P = 40 cm

V = 64 cm3 V = 128 cm3

8 cm4 cm

4 cm 4 cm

4 cm4 cm

If you tripled one dimension, the area would also triple. Now what if both dimensions were doubled?

When both dimensions were doubled, the area quadrupled. In other words, it was doubled twice. The perimeter doubled as well.

We now turn to solids. This time, the length of a cube is doubled. The figure becomes a rectangular prism, and the volume doubles as well.



RemembeR

V = 64 cm3 V = 256 cm3

8 cm4 cm

4 cm4 cm

8 cm

4 cm

V = 512 cm3

8 cm

8 cm

8 cm

If you tripled one dimension, the volume would triple as well. Consider what happens when two dimensions and three dimensions are doubled.

When two dimensions were doubled, the volume quadrupled, or doubled twice. When three dimensions were doubled, the volume became eight times greater, which is the same as being doubled three times.

Surface Area to Volume RatioThe surface area to volume ratio of a figure plays a large role in the sciences, such as in studying the cooling rates of animals. An animal with a greater surface area to volume ratio will lose heat more quickly than an animal with a smaller surface area to volume ratio.

The ratio is found by simply dividing the surface area of a figure by its volume. The surface area to volume ratio of a cube is

surface area of a cube __________________ volume of a cube = 6s2 ___ s3 = 6 __ s

Compare the ratios for cubes with side lengths of 2, 3, and 6:

side length of cube, s 2 3 6

surface area to volume ratio

of a cube, 6 __ s 6 ___ (2) = 3 6 ___ (3) = 2 6 ___ (6) = 1

A ratio is a comparison of two quantities by division.



Notice that as the side length increased, the surface area to volume ratio decreased. This is true for any solid: The surface area to volume ratio is inversely proportional to the size of the solid.

Maximizing Volume and Minimizing Surface AreaA common dilemma in manufacturing is determining how to maximize the volume of a box while minimizing its surface area. In this sketch, the yellow area is a net of a box without a top. You can increase or decrease the length of the sides of the squares in white to change the volume and surface area of the yellow box until you are content with the measurements.

Summary

• Theformulafor finding thesurfaceareaS of a right rectangular prism with length l, width w, and height h is S = 2lw + 2lh + 2wh. The formula for the volume V of a right rectangular prism is V = lwh.

• Acubeisarightrectangularprismwhoseeveryfaceisasquare.Withaside length s, S = 6s2 and V = s3.

• Doublingonedimensionofaplanefiguredoublesthearea;doublingtwodimensions quadruples the area and doubles the perimeter.

• Doublingonedimensionofasolidfiguredoublesthevolume;doublingtwodimensions quadruples the volume; anddoubling all three dimen-sions makes the volume eight times greater.

• Thesurfaceareatovolumeratioofasolidisinverselyproportionaltothesize of the solid.


Surface area and Volume of PriSmS 205

Surface Area and Volume of Prisms

height

lateral face

base

base

height

OBJECTIVESMany buildings are built in the shape of a prism. If an entire building is not a prism, then usually at least a part of it is. Homeowners may calculate surface area when determining how much paint they need to paint the exterior of their home. They may calculate volume when choosing what size air conditioner they need to buy to effectively cool a room. Now you will learn how to calculate the surface area and volume of any prism.

► Define and identify parts of a prism.

► Develop and use the formula for the surface area of a right prism.

► Develop and use Cavalieri’s Principle.

► Find the volume of any prism.

KEYWORDS altitude bases of a prismbase area cross sectionheight lateral arealateral face

Prisms

Recall that a prism is a polyhedron with two parallel, congruent faces called bases. The other faces of the prism are called lateral faces. An altitude of a prism is a perpendicular line segment that joins the planes of the bases. An altitude may lie inside, on, or outside the solid. The height of a prism is the length of an altitude.

Surface Area of a PrismYou know that surface area is found by calculating the sum of the areas of every face in a solid. For a right prism, there is another way. As shown in the following sketches, the lateral faces of any right prism “unfold” to form a rectangle. The length of the base of the rectangle equals the perimeter of the base of the prism and the width of the rectangle equals the height. Lateral area L denotes this area of the rectangle, because it is the sum of the areas of the lateral faces.



These are nets of two right prisms. The lateral areas are shown in blue.

This is the net of a right rectangular prism.

Because the blue areas are rectangles, their areas can be found by multiplying the length, which is the perimeter P of the base, by the width, which is the height h of the prism. The expression is hP. To then find the surface area of the entire prism, add the base areas to the expression. Each base area is B, so 2B is added to the lateral area, hP.

Formula for Surface Area of a Right Prism

The surface area S of a right prism is the sum of the lateral area L and the areas of the two bases 2B.

S = L + 2B

By replacing L with the product of the height h and the perimeter of the base of the prism P, the formula can also be written as

S = hP + 2B

This is the net of a right triangular prism.


Surface area and Volume of PriSmS 207

h = 5 cm

B = 6 cm2

B = 6 cm2 B = 6 cm2

h = 5 cm h = 5 cm

The calculator screen below shows how to use the formula to find the surface area of the right rectangular prism shown.

Volume of a PrismYou learned already that the volume of a rectangular prism can be found by using the formula V = lwh. Because lw gives the area of the base B, the formula also can be written as V = Bh.

The blue regions in the prisms are called cross sections. A cross section is the intersection of a plane and a solid. Because the bases are congruent, every cross section in a prism that is parallel to a base is congruent.

Notice that the areas of the cross sections in all three prisms are the same. The heights are also equal. It stands to reason then that the volumes of all three prisms are equal as well. This is known as Cavalieri’s Principle. Cavalieri’s Principle is true for any two solids, not just two prisms.

Cavalieri’s Principle

If two solids have equal heights and the cross sections formed by every plane parallel to the bases of both solids have equal areas, then the

two solids have equal volume.

Therefore, if the volume of the rectangular prism can be found using the formula V = Bh, then so can the volume of the others—even the oblique prism!

Formula for the Volume of Any Prism

The volume V of a prism where B is the base area and h is the height is V = Bh

2 cm

3 cm

3 cm



3 in.

5 in.

10 in.

3 in.

To find the volume of the right trapezoidal prism shown, first find the area of a base.

B = 1 __ 2 (b1 + b2)h

= 1 __ 2 (5 + 3)3

= 1 __ 2 (24)

= 12

Then, use the volume formula.

V = Bh = (12)(10) = 120

The volume of the right trapezoidal prism is 120 in3.

The screen below shows how to find the volume using a calculator.

Summary

• Analtitudeofaprismisaperpendicularlinesegmentthatjoinstheplanesof the bases. An altitude may lie inside, on, or outside the solid. The height of a prism is the length of an altitude.

• ThesurfaceareaS of a right prism is equal to the lateral area L plus two times the area of a base: S = L + 2B. Because the lateral area equals the height h of the prism times the perimeter P of the base, the formula for surface area can also be written as S = hP + 2B.

• ThevolumeofanyprismV is the area of the base B times the height h: V = Bh.


Surface area and Volume of PyramidS 209

Surface Area and Volume of Pyramids

height

base lateral face

vertex

square pyramid

OBJECTIVES This large Egyptian pyramid is called the Bent Pyramid. Structural problems during its construction caused the builders to change the angle of elevation from 55° to 43° and, in doing so, caused the pyramid to have this unique bent shape.

► Define and identify parts of a pyramid.

► Develop the formula for the surface area of a regular pyramid.

► Find the surface area of regular pyramids.

► Find the volume of pyramids.

KEYWORDS base edge lateral edgepyramid regular pyramidslant height vertex of a pyramid

Pyramids

A pyramid is a polyhedron with a polygonal base and lateral faces. The faces are triangles, and they meet at a common vertex. The common vertex is called the vertex of the pyramid.



s

s s s s

l

l l l l

lateral edge

slant height

base edge

regular hexagonal pyramid

height

slant height

oblique triangular pyramid

Pyramids are classified by the shapes of their bases. So if the base of a pyramid is a square, the pyramid is called a square pyramid. A regular pyramid has a base that is a regular polygon and lateral faces that are congruent isosceles triangles.

The edges that form the base of a pyramid are called base edges. The remaining edges are lateral edges.

The altitude of a pyramid is a perpendicular line segment that joins the vertex to the plane of the base. An altitude may lie inside, on, or outside a pyramid. The height of a pyramid is the length of the altitude. The slant height l is the length of the altitude of a lateral face. A right pyramid’s altitude intersectsthecenterofitsbase;anobliquepyramid’saltitudedoesnot.

Surface Area of a Regular PyramidYou can study the net of a regular pyramid to determine a formula for its lateral area. Here are two possible nets for a square pyramid. The second shows the lateral faces lined up all in a row.

Each of the triangles has an area of 1 __ 2 sl. The sum of the area of all four triangles, which is also called the lateral area L, is four times that:

L = 4 ( 1 __ 2 sl ) The expression can be rearranged:

L = 1 __ 2 l(4s)

Finally, because 4s is the perimeter of the base, it can be replaced with P.

L = 1 __ 2 lP


Surface area and Volume of PyramidS 211

8 cm

7 cm

h

RemembeR

In the sketch below, the measurements in bold show the lateral area of the pyramid to be 4 cm2.Thefirstcalculationisfourtimestheareaofonetriangle;the second calculation uses the formula we derived.

For a square, P = 4s and B = s2

Once you know the lateral area of a pyramid, all you have to do is add on the area of the base B and you will have the total surface area.

Formula for the Surface Area of a Regular Pyramid

The surface area S of a regular pyramid with lateral area L and base area B is

S = L + B

By replacing L with one-half times the product of the slant height l and the perimeter P of the base of the prism, the formula can also be written as

S = 1 __ 2 lP + B

Here is how you can use the formula to find the surface area of the square pyramid below.

S = 1 __ 2 lP + B

=  1 __ 2 7(32) + (8)2

= 112 + 64

= 176

The surface area of the pyramid is 176 cm2.

Volume of a PyramidThis figure shows a square pyramid and a cube. The areas of the bases of each figure, and their heights, are equal. The ratio of the volume of the pyramid to the volume of the cube is 1 __ 3 . Because the volume of the prism can be found using V = Bh, the volume of the pyramid can be found by taking one-third of that product.



9 in.7 in.

4 in.

Formula for the Volume of a Pyramid

The formula for the volume of a pyramid V with base area B and height h is

V = 1 __ 3 Bh

Cavalieri’s Principle tells us that the formula for the volume of a pyramid can be used to find the volume of any pyramid, including oblique pyramids.

It is possible to use the formula to find the volume of the oblique pyramid at left. Begin by finding the area of the base.

B = 1 __ 2 bh

= 1 __ 2 (7)(4)

= 1 __ 2 (28)

= 14

Now, use the volume formula.

V = 1 __ 3 Bh

= 1 __ 3 (14)(9)

= 42 The volume of the pyramid is 42 in3.

Summary

• Apyramidisapolyhedronwithapolygonalbaseandlateralfaces.Thefaces are triangles, and they meet at a common vertex.

• Theslantheightofapyramidisthelengthofthealtitudeofalateralfaceof the pyramid.

• Aregularpyramidhasabasethatisaregularpolygonandlateralfacesthat are congruent isosceles triangles.

• ThelateralareaL of a regular pyramid is one-half of the product of the slant height l of the pyramid and the perimeter P of the base: L = 1 __ 2 lP.

• ThesurfaceareaS of a regular pyramid is the sum of the lateral area L and the area of the base B: S = L + B or S = 1 __ 2 lP + B.

• ThevolumeV of a pyramid is one-third of the product of the area of the base B and the height h of the pyramid: V = 1 __ 3 Bh.


Surface area and Volume of cylinderS 213

Surface Area and Volume of Cylinders

RemembeR

OBJECTIVES Silos are huge cylinders that are used to store food for farm animals. Silos weren’t always cylindrical. They began as rectangular containers, but the walls became bowed after a time from the pressure of the contents. Air then entered the silo, which spoiled the food and caused acids to rot the wood. Toward the end of the nineteenth century, cylindrically shaped silos were designed, which solved these problems.

Farmers need to know how to use the formula for the surface area of a cylinder to repaint a silo. They need the formula for the volume of a cylinder to determine how much storage space they have.

► Define and identify the parts of a cylinder.

► Develop and use the formula for the surface area of a cylinder.

► Develop and use the formula for the volume of a cylinder.

KEYWORDS axis of a cylinder cylinderlateral surface oblique cylinderright cylinder

Cylinders

A cylinder is a solid with two parallel, congruent, circular bases joined by a curved surface. The curved surface of a cylinder is called the lateral surface. An altitude of a cylinder is a perpendicular line segment that joins the planes of the bases. An altitude may lie inside, on, or outside the cylinder. The height

A cylinder is not a polyhedron because its surfaces are not polygons.



height

base

base

axis

lateral surface

axis

height

h

2πr

of a cylinder is the length of an altitude. The line connecting the centers of the bases is the axis of the cylinder. If the axis is an altitude, then the cylinder is a right cylinder. Otherwise it is an oblique cylinder.

Surface Area of a Right CylinderAs with other solids you have studied, the base area of a cylinder is the area of its base and the lateral area is the area of the lateral surface. The surface area of a right cylinder is the sum of the lateral area and two times the base area.

Take a closer look at the lateral surface. As shown below, the lateral surface can be unrolled to form a rectangle. The length of the rectangle is the circumference of the circle, πd, or 2πr. The height of the rectangle is the height of the cylinder. Therefore the lateral area can be written as 2πrh.

right cylinders oblique cylinder

In the sketch below, the radius of the base is 1 centimeter, and the rectangle has a length of 2πr, which is approximately 6.28 centimeters.


Surface area and Volume of cylinderS 215

4 cm

8 cm

Now that you know that the area of the lateral surface is 2πrh, all you have to do to find the surface area is to add the base areas. Each base is a circle, so the area of one base is πr2 and the area of both bases is 2πr2.

Formula for the Surface Area of a Right Cylinder

The surface area S of a right cylinder with lateral area L and base area B is S = L + 2B

By replacing L with 2πrh, and B with πr2, you can write the formula asS = 2πrh + 2πr2

Now use the formula to find the surface area of this cylinder.

S = 2πrh + 2πr2

= 2π(4)(8) + 2π(4)2

= 64π + 32π = 96π ≈ 302

The surface area of the cylinder is approximately 302 cm2.

Most calculators have a key for π, although you may need to press another key first. The answer is still an estimate because it has been rounded to fit on the display, but it is a better estimate than using 3.14. Using a precise approximation for π allows us to round our answers to the tenths, hundredths, and even thousandths place with confidence. As shown in the problem above, always wait until the last possible step before rounding.

Volume of a CylinderYou can use Cavalieri’s Principle to find the volume of a cylinder because the principle refers to solids, not just to prisms. A cylinder with the same height and cross-sectional areas at every level as a prism will have the same volume as the prism, which can be found using V = Bh.

Formula for the Volume of a Cylinder

The formula for the volume V of a cylinder with base area B and height h is V = Bh

By replacing B with πr2, you can write the formula asV = πr2h



10 cm

12 cm

Cavalieri’s Principle tells us that the formula for the volume of a cylinder can be used to find the volume of any cylinder, including oblique cylinders.

To find the volume of the cylinder shown at left, substitute 10 for r and 12 for h. V = πr2h = π(10)2(12) = 1200π ≈ 3770 The volume of the cylinder is approximately 3770 cm3.

Summary

• Acylinderisasolidwithtwoparallel,congruent,circularbasesjoinedbya lateral surface.

• Thelineconnectingthecentersofthebasesofacylinderistheaxis.Iftheaxis is an altitude, then the cylinder is a right cylinder.

• Thelateralareaofarightcylinderis2πrh, where r is the radius of the bases and h is the height of the cylinder.

• ThesurfaceareaofarightcylinderisL + 2B, where L is the lateral area and B is the area of a base. The formula for the surface area can be written then as S = 2πrh + 2πr2.

• ThevolumeofacylinderisBh, where B is the area of a base and h is the height of the cylinder. When B is replaced with the expression for the area of a circle, the formula for the volume of a cylinder is written as V = πr2h.


Surface area and Volume of coneS 217

Surface Area and Volume of Cones

height

vertex

base

slant height

lateral surface

axis

height

axis

right cones oblique cone

OBJECTIVES Many people have claimed to have invented the ice cream cone. What is certain, however, is that after its appearance at the 1904 St. Louis World’s Fair, the ice cream cone became extremely popular in the United States. Prior to that time, people ate ice cream out of a dish or a paper cone. New businesses were born, and cone production reached 245,000,000 in 1924. Today machines can produce 150,000 cones a day. Have you ever wondered how much batter it takes to make each cone? How much ice cream does each cone hold? You can answer these types of questions once you understand how to calculate surface area and volume.

KEYWORDS cone oblique coneright cone sectorslant height of a cone vertex of a cone

Cones A cone is a solid with a circular base, a vertex, and a curved surface. The curved surface of a cone is called the lateral surface. The altitude of a cone is a perpendicular line segment that joins the vertex to the plane of the base. An altitude may lie inside, on, or outside a cone. The height of a cone is the length of the altitude. The slant height of a cone is the distance from the vertex to a point on the edge of the base. The line connecting the vertex of a cone to the center of the base is the axis of the cone. If the axis is an altitude, then the cone is a right cone. Otherwise it is an oblique cone.

► Define and identify types of cones and their parts.

► Develop and use the formula for the surface area of a right cone.

► Develop and use the formula for the volume of a cone.

right cones oblique cone



l

l

2πr

10 ft

5 ft

RemembeR

Surface Area of a Right ConeThe surface area of a right cone is the sum of the lateral area and the base area.

The lateral surface of a right cone can be unrolled to form the fanlike shape shown as a shaded area. The length of the arc is the circumference, or perim-eter, of the base of the cone. The formula for circumference is 2πr. The sides are the slant height l of the cone. Notice that the fanlike shape resembles a triangle. The lateral surface area of a cone is similar to the area of a triangle 1 __ 2 bh, or 1 __ 2 (2πr)(l), which simplifies to πrl.

You can check this conjecture algebraically. The diagram shows that the lateral surface is a sector of a circle where the slant height is the radius. Recall that a sector is the region bounded by two radii and part of the circle. The area of a sector is proportional to the area of its circle. You can write and solve the following proportion to find the area of the sector.

area of sector ___________ area of circle = length of arc

___________________ circumference of circle

x ___ πl 2

=  2πr ____ 2πl

2πlx = 2π2rl 2

x =  2π2rl 2 _____ 2πl = πrl

Now that you know that the lateral area of a right cone is πrl, all you need to do to find the surface area is add the area of the base. The base is a circle, so the base area is πr2.

Formula for the Surface Area of a Right Cone

The surface area S of a right cone with lateral area L and base area B is S = L + B

By replacing L with πrl, and B with πr2, the formula can also be written as S = πrl + πr2

Use the formula to find the surface area of the right cone.

S = πrl + πr2

= π(5)(10) + π(5)2

= 50π + 25π = 75π ≈ 236

The surface area of the cone is approximately 236 ft2.

You can solve a proportion by “cross-multiplying.”


Surface area and Volume of coneS 219

hh

9 in.

8 in.

Formula for the Volume of a Cone

The formula for the volume V of a cone with base area B and height h is

V = 1 __ 3 Bh

When B is replaced with πr2, the formula can also be written as

V = 1 __ 3 πr2h

Cavalieri’s Principle tells you that the formula for the volume of a cone can be used to find the volume of any cone, including oblique cones.

Find the volume of the cone shown.

V = 1 __ 3 πr2h

= 1 __ 3 π(8)2(9)

= 192π

≈ 603

The volume of the cone is approximately 603 in3.

On a calculator, you can divide by three instead of multiplying by one-third. This can be done in one step or two.

Volume of a ConeWe can use Cavalieri’s Principle to find the volume of a cone. A cone with the same height and cross-sectional areas at every level as a pyramid will have the same volume as the pyramid, which is found by using V = 1 __ 3 Bh.

or



Summary

• Aconeisasolidwithacircularbase,avertex,andacurvedsurface.• Thelineconnectingthevertextothecenterofthebaseistheaxis.Ifthe

axis is an altitude then the cone is a right cone.• Thelateralareaofarightconeisπrl where r is the radius of the base and

l is the slant height of the cone. The base of a cone is a circle with area πr2.

• ThesurfaceareaS of a right cone can be found by adding the base area to the lateral area, which results in the formula S = πrl + πr2.

• ThevolumeV of a cone can be found with the formula V = 1 __ 3 πr2h.


Surface area and Volume of SPHereS 221

Surface Area and Volume of Spheres

center

radius

diameter

OBJECTIVES The surface of a modern soccer ball is made of 20 hexagonal and 12 pentago-nal surfaces forming an almost perfect sphere. The formal name of this figure is truncated icosahedron. Its more common name is buckyball, named after architect and engineer Buckminster Fuller. He created this design and many variations of it during his long career.

Manufacturers of soccer balls calculate surface area to know how much leather they will need.

► Define and identify parts of a sphere.

► Develop and use the formula for the volume of a sphere.

► Develop and use the formula for the surface area of a sphere.

KEYWORDS annulus sphere

Spheres

A sphere is the set of all points in space that are a given distance from a point called the center. The radius of a sphere is a line segment joining the center of the sphere and a point on the surface of the sphere.

The diameter is a line segment passing through the center that joins two points on the sphere.



h = 2r h = 2r

r

rx

x

xr

y

x

r

Volume of a SphereWhen a plane and a sphere intersect at more than one point, their intersection creates a circle. This figure shows a plane intersecting a sphere at a distance x from the center of the sphere. The radius of the sphere is r. The purple circle created by the intersection has a radius y. You know from the Pythagorean Theorem that x2 + y2 = r2.

You can solve this equation for y2: y2 = r2 – x2

The formula for the area of the purple circle is A = πy2. Substitute r2 – x2 for y2 and distribute. A = πy2 = π(r2 – x2) = πr2 – πx2

The expression πr2 – πx2 can be viewed as the difference between the areas of two circles, one with radius r and one with radius x. In the diagram, there are two concentric circles. The difference is the darker area, which is called the annulus. The annulus is the region between two concentric circles.

The area of the purple circle in the sphere has the same area as the annu-lus. To use Cavalieri’s Principle, you will need to find a solid that has cross sections like the concentric circles in the diagram—and it needs to work for every cross section of the sphere.

You can use part of a cylinder. The cylinder in the diagram has two identi-cal cones inside it. The bases of the cones are the bases of the cylinder, and the vertices meet in the center of the cylinder. The part of the cylinder you want is the part that remains when the two cones are removed.

Both the cylinder and the sphere have a height of 2r. The area of every cross section of the cylinder is equal to every area of the cross section of the sphere at that height. In the diagram above, visualize moving the cross sec-tions up and down the figures, and match the size of the annulus on the left with the corresponding circle on the right—it’s larger near the center of the sphere, and smaller near the top and bottom of the sphere.


Surface area and Volume of SPHereS 223

B

r

6 in.

By Cavalieri’s Principle, the two solids have equal volume. To find the volume of the sphere, subtract two cones from the cylinder.

Volume of a Sphere = Volume of a Cylinder – 2(Volume of a Cone)

= πr2h – 2 ( 1 __ 3 πr2h ) = πr2(2r) – 2 ( 1 __ 3 πr2r ) = 2πr3 – 2 __ 3 πr3

= 4 __ 3 πr3

Formula for the Volume of a Sphere

The formula for the volume V of a sphere with radius r is

V = 4 __ 3 πr3

To find the volume of the sphere shown, substitute 6 for r.

V = 4 __ 3 πr3

= 4 __ 3 π(6)3

= 288π ≈ 905

The volume of the sphere is approximately 905 in3.

You can use a calculator to find the volume of a sphere as shown above.

Surface Area of a SphereIn the introduction you learned that a soccer ball closely resembled a sphere. It’s not a sphere because the surface is not perfectly smooth—but what if the polygons on the surface were much, much smaller? What if they were so small you almost couldn’t see them? As the polygons on the surface got smaller, the ball would more closely resemble a true sphere.

Now imagine covering the surface of a sphere with tiny congruent poly-gons. If a line segment were to join each polygon’s vertex with the center of the sphere, the sphere could actually be viewed as a collection of tiny congru-ent pyramids. The height of each pyramid would be the radius of the sphere. Therefore, the volume of each pyramid would be 1 __ 3 Br. You could write the volume of the sphere as the sum of the volumes of all the pyramids:

V = 1 __ 3 Br + 1 __ 3 Br + 1 __ 3 Br + …



4 cm

With n pyramids, you could write it more neatly like this:

V = n 1 __ 3 Br

You could also rearrange the equation to be

V = 1 __ 3 r (nB)

This is convenient because nB is the total area of the bases of the pyramids that form the surface area S of the sphere.

V = 1 __ 3 r (S)

Use algebra to solve the equation for S.

S = 3V ___ r

Now substitute the formula for the volume of a sphere for V and simplify.

S = 3 ( 4 __ 3 πr3 )

________ r = 4πr2

Formula for the Surface Area of a Sphere

The formula for the surface area S of a sphere with radius r is S = 4πr2

To find the surface area of the sphere shown, substitute 4 for r.

S = 4πr2

= 4π(4)2

= 64π ≈ 201

The surface area of the sphere is approximately 201 cm2.

Summary

• Asphereisthesetofallpointsinaspacethatareagivendistancefromapoint called the center. The line segment joining the center of the sphere to any point on the edge of the sphere is called the radius.

• Theannulusistheregionbetweentwoconcentriccircles.• TheformulaforthevolumeV of a sphere with radius r is V = 4 __ 3 πr3.• TheformulaforthesurfaceareaS of a sphere with radius r is S = 4πr2.


tHree-dimenSional Symmetry 225

Three-Dimensional Symmetry

RECONNECT TO THE BIG IDEA

OBJECTIVES A torus is a doughnut-shaped figure. It is created by rotating a circle about a line. This torus was created by rotating a circle about a vertical line.

The size of the “doughnut hole” depends upon how far away the circle is placed from the line it is rotated about. When the circle is placed so that the diameter of the circle is on the line, a sphere is created instead.

► Perform reflections and translations in three-dimensional space.

► Visualize, recognize, and sketch solids of revolution.

► Identify planes of symmetry for solid figures.

KEYWORDS axis of revolutionplane of symmetrysolid of revolution

Transformations in Three Dimensions

Recall that a transformation is a one-to-one mapping between two sets of points. In three-dimensional space, points, lines, two-dimensional figures, and three-dimensional solids can be reflected across a plane, translated along a line, or rotated about an axis. As in two-dimensional transformations, transformations in three dimensions preserve the size and shape of the original figure.

Remember Isometric transformations—translations, rotations, reflections—do not change distances, angles, or areas.



(–4, 4, 3)

(–5, 0, 6)

(4, 4, 3)

(5, 0, 6)

z

y

x

y

z

x

(0, 5, 4)

(2, 0, 0)

(2, 0, 2)

(0, 0, 2)(0, 2, 2)

(2, 2, 0)

(0, 0, 0)(2, 2, 2)

(0, 2, 0)

(0, 7, 4)

(–2, 7, 4)(0, 7, 6)

(–2, 7, 6)

(–2, 5, 4)

(–2, 5, 6)

(0, 5, 6)

Reflections In two dimensions, a figure can be reflected across a line. In three dimensions, a figure can be reflected across a plane. To reflect across planes formed by the intersection of the axes, use the following ordered-triple rules.

to reflect across the... use... xy-plane (x, y, z)→(x, y, –z) xz-plane (x, y, z)→(x, –y, z) yz-plane (x, y, z)→(–x, y, z)

For example, to reflect the blue line segment below across the yz-plane, use the ordered-triple rule (x, y, z)→(–x, y, z).

(5,0,6)→(–5,0,6) (4,4,3)→(–4,4,3)

Translations To translate a solid in space, add or subtract an amount to or from each coordinate.

For example, you can translate the blue cube two units back, five units right, and four units up with the ordered-triple rule (x, y, z)→(x – 2, y + 5, z + 4).

(2,0,0)→(0,5,4) (2,2,0)→(0,7,4)(0,2,0)→(–2,7,4) (0,0,0)→(–2,5,4)(2,0,2)→(0,5,6) (2,2,2)→(0,7,6)(0,2,2)→(–2,7,6)(0,0,2)→(–2,5,6)



A B

CD

D

FE

A B

Solids of RevolutionA solid of revolution, sometimes called a solid of rotation, is a three-dimensional solid that can be generated by rotating a plane figure about a line that lies in the same plane as the figure. The line is called the axis of revolution.

You can generate this cylinder by rotating rectangle ABCD about line AD. ___ AD is the axis of the cylinder, and because it is a right cylinder, AD is the

height. ___

BC is the side of the rectangle that creates the lateral surface. ___

AB and ___ DC generate the bases of the cylinder.

You can generate this cone by rotating △DEF about line DE. ___

DE is the axis of the cone, and because it is a right cone, DE is the height.

___ DF is the

side of the triangle that creates the lateral surface. ___

EF generates the base of the cone.

The cylinder and cone were created with polygons, but you can rotate other shapes around a line as well. If you rotated a semicircle about line AB, you would form a sphere.

Planes of SymmetryJust as a line of symmetry in a two-dimensional figure divides the figure into two parts that are mirror images of each other, a plane of symmetry divides a three-dimensional figure into two congruent parts.

A tetrahedron is a solid made up of four congruent triangular faces. It has six planes of symmetry. Each plane contains one of the six edges and bisects the opposite face. One plane of symmetry is shown below.



line of symmetry not a line of symmetry

A cube has nine planes of symmetry. Three are shown below.

The tetrahedron and cube have a finite number of planes of symmetry. However, solids created by rotating a plane figure about a line have an infinite number of planes of symmetry that contain the axis of revolution. They may also have planes of symmetry that do not contain the axis of revolution.

Let’s look back at the torus. Along with the infinite number of planes that contain the axis, it also has a plane of symmetry perpendicular to the axis of revolution, as shown in the far right illustration.

The torus has a plane of symmetry perpendicular to the axis of revolution because the plane figure that was rotated to create it, a circle, has a horizontal line of symmetry. Because a right triangle does not have a horizontal line of symmetry, a cone only has planes of symmetry that contain the axis of revolution.



Summary

• Just like two-dimensional transformations, transformations in threedimensions preserve the size and shape of the original figure.

• Asolidofrevolutionisathree-dimensionalsolidthatcanbegeneratedbyrotating a plane figure about a line that lies in the same plane as the figure. Cylinders, cones, and spheres are solids of revolution.

• Aplaneofsymmetrydividesathree-dimensionalfigureintotwocongruentparts.

• Solidscreatedbyrotatingaplaneaboutalinehaveaninfinitenumberofplanes of symmetry.


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